arXiv subjects

Jujie Wu

Publications and source records attributed to Jujie Wu.

15 recordsLinked to original sources

arXiv2026

Energy asymptotics of holomorphic functions with application to Calder\'{o}n-Zygmund theory in $\mathbb{C}$

The Calder\'on-Zygmund theory establishes the boundedness of singular integral operators on $L^p$ spaces for $1 < p < \infty$, yet it encounters a failure at the endpoint $p = 1$. While radial counterexamples in $\mathbb{R}^n$ are well-documented, Pan-Shao-Wang-Wu \cite{psww2026} has showed that every nonconstant holomorphic function provides a counterexample to the Poisson equation within the Calder\'on-Zygmund framework, with the singular locus being a complex subvariety of codimension one. In this paper, we focus on the complex one-dimensional case and establish stronger results. We prove asymptotic formulas with explicit constants for both the level-set integral and the sublevel-set energy. Then we give simplified proofs of the universal counterexamples to Calder\'on-Zygmund theory at $p = 1$ in $\mathbb{C}$. Additionally, we construct a new family of counterexamples at the endpoint $p = \infty$, showing that the failure of $W^{2,\infty}$-regularity is also a universal phenomenon in complex one dimension.

math.CV↗

arXiv2026

Boundary zeros of stable polynomials in the unit ball

Interpolation theory in the unit ball and semi-algebraic geometry yield explicit descriptions of the boundary zeros of stable polynomials. Given a polynomial $p\in \mathbb{C}[z_1, ...,z_n]$ that is zero-free in the unit ball and vanishes on the sphere along submanifolds of dimension at most one, we describe the boundary zeros $\mathcal{Z}(p)\cap\mathbb{S}_n$ in terms of peak sets for $A^\infty(\mathbb{B}_n)$. In particular, in the setting $n=2$, we achieve a characterization by proving that every accumulation point of $\mathcal{Z}(p)\cap\mathbb{S}_2$ lies in the relative interior of an one dimensional real analytic submanifold, and that these submanifolds form a foliation of the non-isolated part of $\mathcal{Z}(p)\cap\mathbb{S}_2$. As an application of the developed theory, we obtain a characterization of cyclic polynomials without weak essential singularities in the Dirichlet-type space $\mathcal{D}_{n-1/2}(\mathbb{B}_n)$. A theory for more general geometric settings of the boundary zeros is also developed, aiming to provide a starting point for further extensions.

math.CV↗

arXiv2026

Riesz $\alpha$-capacity of Cantor sets and cyclicity in Dirichlet-type spaces

We examine the threshold of the cyclicity for functions in Dirichlet-type spaces $\mathcal{D}_{\alpha}$, $\alpha\in(0,1]$. Given a fixed $\alpha^{*}\in(0,1]$, we construct a holomorphic function $f\in\mathcal{D}_{\alpha^{*}}$ which is cyclic in $\mathcal{D}_{\alpha}$ for all $\alpha<\alpha^{*}$, but fails to be cyclic in $\mathcal{D}_{\alpha^{*}}$. This function serves as a counterexample to the persistence of cyclicity at the critical index $\alpha^{*}$. Throughout the construction process, we work with generalized Cantor sets and study their Riesz $\alpha$-capacity.

math.CV↗

arXiv2026

Energy estimates for level sets of holomorphic functions and universal counterexamples to Calder\'on-Zygmund theory

We demonstrate that the failure of $L^1$ regularity in Calder\'on-Zygmund theory is a universal phenomenon: every non-constant holomorphic function in $\C^n$ generates a counterexample to the Poisson equation. In order to achieve this goal, we shall establish sharp level-set estimates that link harmonic analysis to the geometry of complex structure through Hironaka's resolution of singularities and the \L{}ojasiewicz gradient inequality.

math.CV↗

arXiv2025

Equivalence between VMO functions and plurisubharmonic functions with zero Lelong numbers

We prove that a plurisubharmonic function on a domain in the complex Euclidean space is a locally VMO (Vanishing Mean Oscillation) function if and only if its Lelong number at each point vanishes. We also give a global version of this result when the boundary of the domain satisfies the \textit{interior sphere condition}. An example emphasizes the importance of this condition. These equivalences contribute to a better understanding of the behavior of singular plurisubharmonic functions. We end the paper by discussing the link between the residual Monge-Ampère mass and VMO functions, by providing examples.

math.CV↗

arXiv2025

The Grothendieck Theorem in Bergman Spaces

In this paper, we prove that if $E$ is a closed subspace of the holomorphic $L^p$-integrable space and is also contained in the holomorphic $L^q$-integrable space, for any $p > 1$ and any $q > p$, then the dimension of $E$ must be finite.

math.CV↗

arXiv2024

Holomorphic motion, rational approximation and an equivalent formulation of the Riemann Hypothesis

A compact subset $K$ of the complex plane $\C$ is a set of polynomial (respectively rational) approximation if $P(K)=A(K)$ (respectively $R(K)=A(K)$), where $P(K)$ (respectively $R(K)$) is the family of functions on $K$ which are uniform limits of polynomials (respectively rational functions, having no poles on $K$) and $A(K)$ is the family of continuous functions on $K,$ which are holomorphic on the interior of $K.$ In the class of compact sets, the property of being a set of polynomial approximation is easily seen to be invariant under holomorphic motion. We show that this is no longer the case for rational approximation. Secondly, we show that the Riemann Hypothesis holds if and only if a certain map is a holomorphic motion.

math.CV↗

arXiv2024

Density in weighted Bergman spaces and Bergman completeness of Hartogs domains

We study the density of functions which are holomorphic in a neighbourhood of the closure $\overlineΩ$ of a bounded non-smooth pseudoconvex domain $Ω$, in the Bergman space $ H^2(Ω,φ)$ with a plurisubharmonic weight $φ$. As an application, we show that the Hartogs domain $$ Ω_α: = \{(z,w) \in D\times \C: |w|< δ^α_D(z) \}, \ \ \ α>0, $$ where $D\subset \subset \C$ and $δ_D$ denotes the boundary distance, is Bergman complete if and only if every boundary point of $D$ is non-isolated.

math.CV↗

arXiv2021

Boundary behavior of the Szegö kernel

We give a Hörmander-type localization principle for the Szegö kernel $S_Ω(z)$. We also show that for each boundary point $z_0$, $S_Ω(z)\gtrsim|z-z_0|^{-\frac{1}{3}}$ holds non-tangentially for any bounded pseudoconvex domain with smooth boundary in ${\mathbb C}^2$.

math.CV↗

arXiv2020

Poincare Series And Very Ampleness Criterion For Pluri-canonical Bundles

Let $X$ be a compact quotient of a bounded domain in $\mathbb C^n$. Let $K_X$ be the canonical line bundle of $X$. In this paper, we shall introduce the notion of $S$ very ampleness for the pluri-canonical line bundles $mK_X$ by using the Poincaré series. The main result is an effective Seshadri constant criterion of $S$ very ampleness for $mK_X$. An elementary proof of surjectivity of the Poincaré map is also given.

math.CV↗

arXiv2020

Weighted $L^2$ version of Mergelyan and Carleman approximation

We study the density of polynomials in $H^2(E,φ)$, the space of square integrable functions with respect to $e^{-φ}dm$ and holomorphic on the interior of $E$ in $\mathbb{C}$, where $φ$ is a subharmonic function and $dm$ is a measure on $E$. We give a result where $E$ is the union of a Lipschitz graph and a Carathéodory domain, that we state as a weighted $L^2$-version of the Mergelyan theorem. We also prove a weighted $L^2$-version of the Carleman theorem. Keywords: Mergelyan theorem, Carleman theorem, Weighted $L^2$- spaces, Rectifiable non-Lipschitz arc

math.CV↗

arXiv2019

Weighted-$L^2$ polynomial approximation in $\mathbb{C}$

We study the density of polynomials in $H^2(Ω,e^{-φ})$, the space of square integrable holomorphic functions in a bounded domain $Ω$ in $\mathbb{C}$, where $φ$ is a subharmonic function. In particular, we prove that the density holds in Carathéodory domains for any subharmonic function $φ$ in a neighborhood of $\overlineΩ$. In non-Carathéodory domains, we prove that the density depends on the weight function, giving examples.

math.CV↗

arXiv2018

Weighted approximation in $\mathbb{C}$

We prove that if $\{ φ_j\}_j$ is a sequence of subharmonic functions which are increasing to some subharmonic function $φ$ in $\mathbb{C}$, then the union of all the weighted Hilbert spaces $H(φ_j)$ is dense in the weighted Hilbert space $H(φ)$.

math.CV↗

arXiv2016

A global approximation result by Al Taylor and the strong openness conjecture in C^n

We improve a global approximation result by Al Taylor in C^n for holomorphic functions in weighted Hilbert spaces. The main tools are a variation of the theorem of Hormander on weighted L^2-estimates for the dbar-equation together with the solution of the strong openness conjecture. A counterexample to a global strong openness conjecture in Cn is also given here.

math.CV↗

arXiv2014

Ohsawa-Takegoshi type theorem and extension of plurisubharmonic functions

We prove a Thullen type extension theorem of plurisubharmonic functions across a closed complete pluripolar set, which generalizes a theorem of Siu. Our approach depends on an Ohsawa-Takegoshi type extension theorem for a single point in a bounded complete Kähler domain, which is of independent interest.

math.CV↗